Abstract

This course will aim to give a glimpse of how methods from algebraic combinatorics can be used to analyse the structure of stochastic models which belong in the Kardar-Parisi-Zhang universality class. Models in this class include interacting particle systems, random polymers, growth models and exhibit non gaussian, cubic-root scalings and fluctuations related to models of random matrices.

 The course will start by introducing some of these models and set up the framework of the KPZ scaling limit conjecture. We will then  describe a fundamental combinatorial tool, the Robinson-Schensted-Knuth correspondence,which played a crucial role in establishing the KPZ conjecture for certain models in "zero temperature", starting from the work of Baik-Deift-Johansson. We will subsequently describe the "geometric lifting" of this correspondence due to A.N.Kirillov and the crucial role this played more recently in the analysis of models in "positive temperature".

 Even though the main taste of the course will lie in probability, it will also aim to be appealing to mathematicians with interest in combinatorics, representation theory, integrable systems and analytic number theory. Some familiarity with the basic concepts of Markov processes would be desirable.